Pre-Calculus – Complete Formula Sheet

Functions: Domain & Range

Key Definitions

Domain: All possible input x-values

Range: All possible output f(x)-values

Function Notation

$f: A \to B$ (map from A to B)

$f(x) = $ the output value

$(f \circ g)(x) = f(g(x))$ (composition)

Even & Odd Functions

Even: $f(-x) = f(x)$ (y-axis symmetry)

Odd: $f(-x) = -f(x)$ (origin symmetry)

Common Domains

Polynomial: All reals $\mathbb{R}$

Rational: Exclude zeros of denominator

Square root: $x \geq 0$

Logarithm: $x > 0$

Use interval notation: $[a,b]$ (closed), $(a,b)$ (open), $[a,b)$ (mixed)

Function Transformations

Vertical & Horizontal Shifts

$f(x) + k$: Shift UP k units

$f(x) - k$: Shift DOWN k units

$f(x - h)$: Shift RIGHT h units

$f(x + h)$: Shift LEFT h units

Stretches & Reflections

$af(x)$: Vertical stretch if $a > 1$

$\frac{1}{a}f(x)$: Vertical shrink

$f(bx)$: Horizontal shrink if $b > 1$

$f(\frac{x}{b})$: Horizontal stretch

Reflections

$-f(x)$: Reflect over x-axis

$f(-x)$: Reflect over y-axis

Combined form: $y = af(b(x-h))+k$

Inverse Functions

Conditions for Inverse

Function must be ONE-TO-ONE

Horizontal line test: Each y-value appears once

Inverse Properties

$f(f^{-1}(x)) = x$ (cancel property)

$f^{-1}(f(x)) = x$

Domain of $f^{-1}$ = Range of $f$

Graph: Reflect $y = f(x)$ over $y = x$

Finding Inverse

1. Replace $f(x)$ with $y$

2. Swap $x$ and $y$

3. Solve for $y$

Common Inverses

$f(x) = ax+b \Rightarrow f^{-1}(x) = \frac{x-b}{a}$

$f(x) = x^2 \Rightarrow f^{-1}(x) = \sqrt{x}$ (for $x \geq 0$)

Polynomial Functions

General Form

$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Degree = highest power

Zeros & Factors

$x = a$ is a zero if $P(a) = 0$

$(x-a)$ is a factor

Degree $n$: max $n$ real zeros

End Behavior

$a > 0, n$ even: $\to \infty$ both sides

$a > 0, n$ odd: down left, up right

$a < 0, n$ even: $\to -\infty$ both sides

$a < 0, n$ odd: up left, down right

Graphing Tips

Find zeros (x-intercepts)

Evaluate P(0) for y-intercept

Use end behavior

Special Forms

$f(x) = a(x-r_1)(x-r_2)\cdots$ (factored)

$f(x) = a(x-h)^2+k$ (vertex form)

Rational Functions

Definition

$f(x) = \frac{P(x)}{Q(x)}$ where P, Q are polynomials

Domain: $x$ where $Q(x) \neq 0$

Asymptotes

Vertical: $x = a$ if $Q(a)=0$, $P(a) \neq 0$

Horizontal: Compare degrees of P and Q

Deg(P) < Deg(Q): $y = 0$

Deg(P) = Deg(Q): $y = \frac{a_n}{b_n}$

Deg(P) > Deg(Q): No horizontal asymptote

Holes

If $(x-a)$ cancels from P and Q: hole at $x=a$

Find y-value by substituting into simplified form

Oblique Asymptote

When Deg(P) = Deg(Q) + 1

Use polynomial division to find $y = mx + b$

Exponential Functions

General Form

$f(x) = a \cdot b^x$ (base $b > 0, b \neq 1$)

$f(x) = ae^{kx}$ (natural exponential)

Properties

Domain: All reals $\mathbb{R}$

Range: $(0, \infty)$ if $a > 0$

$b^x > 0$ for all $x$

Horizontal asymptote: $y = 0$ (if $a > 0$)

Growth & Decay

$b > 1$: Growth (rises)

$0 < b < 1$: Decay (falls)

$e \approx 2.71828$ (natural base)

Exponent Laws

$b^x \cdot b^y = b^{x+y}$

$\frac{b^x}{b^y} = b^{x-y}$

$(b^x)^y = b^{xy}$

$(ab)^x = a^xb^x$

$e^{ix} = \cos x + i\sin x$ (Euler's formula)

Logarithmic Functions

Inverse of Exponential

If $y = b^x$ then $x = \log_b y$

$\log_b(b^x) = x$ and $b^{\log_b x} = x$

Definition

$y = \log_b x \Leftrightarrow x = b^y$

Domain: $(0, \infty)$

Range: All reals $\mathbb{R}$

Common Logarithms

$\log x = \log_{10} x$ (common)

$\ln x = \log_e x$ (natural)

Laws of Logarithms

$\log_b(xy) = \log_b x + \log_b y$

$\log_b(\frac{x}{y}) = \log_b x - \log_b y$

$\log_b(x^n) = n\log_b x$

Change of Base

$\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}$

Special Values

$\log_b 1 = 0$

$\log_b b = 1$

Trigonometric Functions

Unit Circle (r = 1)

$\sin\theta = \frac{y}{r}$ (y-coordinate)

$\cos\theta = \frac{x}{r}$ (x-coordinate)

$\tan\theta = \frac{y}{x}$

Reciprocal Functions

$\csc\theta = \frac{1}{\sin\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta}$

Key Angles

$\theta$	$\sin$	$\cos$	$\tan$
$0°$	0	1	0
$30°$	1/2	√3/2	1/√3
$45°$	√2/2	√2/2	1
$60°$	√3/2	1/2	√3
$90°$	1	0	undef

Periods & Amplitudes

$\sin x, \cos x$: Period $2\pi$

$\tan x$: Period $\pi$

Inverse Trig Functions

Definitions

$y = \sin^{-1}(x) \Leftrightarrow x = \sin(y)$

$y = \cos^{-1}(x) \Leftrightarrow x = \cos(y)$

$y = \tan^{-1}(x) \Leftrightarrow x = \tan(y)$

Ranges (Restricted)

$\sin^{-1}$: $[-\frac{\pi}{2}, \frac{\pi}{2}]$ or $[-90°, 90°]$

$\cos^{-1}$: $[0, \pi]$ or $[0°, 180°]$

$\tan^{-1}$: $(-\frac{\pi}{2}, \frac{\pi}{2})$ or $(-90°, 90°)$

Key Properties

$\sin(\sin^{-1}(x)) = x$ for $|x| \leq 1$

$\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$

Identities

$\tan^{-1}(x) = \sin^{-1}(\frac{x}{\sqrt{1+x^2}})$

$\sin^{-1}(-x) = -\sin^{-1}(x)$ (odd)

$\cos^{-1}(-x) = \pi - \cos^{-1}(x)$

Trig Identities

Pythagorean Identities

$\sin^2\theta + \cos^2\theta = 1$

$1 + \tan^2\theta = \sec^2\theta$

$1 + \cot^2\theta = \csc^2\theta$

Sum/Difference Formulas

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle Formulas

$\sin(2\theta) = 2\sin\theta\cos\theta$

$\cos(2\theta) = \cos^2\theta - \sin^2\theta$

$\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

$\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}$

Half Angle Formulas

$\sin^2(\frac{\theta}{2}) = \frac{1-\cos\theta}{2}$

$\cos^2(\frac{\theta}{2}) = \frac{1+\cos\theta}{2}$

Product-to-Sum

$\sin A \sin B = \frac{1}{2}[\cos(A-B)-\cos(A+B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A-B)+\cos(A+B)]$

Solving Trig Equations

General Solutions

$\sin\theta = a \Rightarrow \theta = \sin^{-1}(a) + 2\pi k$ or $\theta = \pi - \sin^{-1}(a) + 2\pi k$

$\cos\theta = a \Rightarrow \theta = \pm\cos^{-1}(a) + 2\pi k$

$\tan\theta = a \Rightarrow \theta = \tan^{-1}(a) + \pi k$

Strategy

1. Isolate the trig function

2. Find reference angle

3. Determine all angles in specified interval

4. Use periodicity for general solution

Common Equations

$\sin\theta = 0$: $\theta = \pi k$ (all integers)

$\cos\theta = 0$: $\theta = \frac{\pi}{2} + \pi k$

$\tan\theta = 0$: $\theta = \pi k$

Factoring Method

$\sin^2\theta - \frac{1}{4} = 0$

$(\sin\theta - \frac{1}{2})(\sin\theta + \frac{1}{2}) = 0$

Always check for extraneous solutions when simplifying!

Law of Sines & Cosines

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

$R$ = circumradius of triangle

When to Use Law of Sines

AAS or ASA (angle-angle-side)

SSA (ambiguous case - may have 2 solutions)

Ambiguous Case (SSA)

If $a < b\sin A$: No solution

If $a = b\sin A$: One solution (right triangle)

If $b\sin A < a < b$: Two solutions

If $a \geq b$: One solution

Law of Cosines

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

When to Use Law of Cosines

SAS (side-angle-side)

SSS (side-side-side)

Area Formula

$K = \frac{1}{2}ab\sin C$

$K = \sqrt{s(s-a)(s-b)(s-c)}$ (Heron's)

where $s = \frac{a+b+c}{2}$ (semi-perimeter)

Polar Coordinates

Conversion

$(r, \theta)$ where $r$ = distance, $\theta$ = angle

$x = r\cos\theta$, $y = r\sin\theta$

$r = \sqrt{x^2 + y^2}$, $\tan\theta = \frac{y}{x}$

Polar Equations

Line through pole: $\theta = \alpha$

Circle: $r = a$ (center at pole)

$r = a\cos\theta$ or $r = a\sin\theta$ (circles)

Common Polar Curves

Rose: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$

Spiral: $r = a\theta$ (Archimedean)

Cardioid: $r = a(1 \pm \cos\theta)$

Lemniscate: $r^2 = a^2\cos(2\theta)$

Arc Length

$L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} \, d\theta$

Area

$A = \frac{1}{2}\int_{\alpha}^{\beta} r^2 \, d\theta$

Multiple representations: $(r,\theta) = (-r, \theta+\pi) = (r, \theta+2\pi k)$

Parametric Equations

Definition

$x = f(t)$, $y = g(t)$ where $t$ is parameter

Eliminates $t$ to get Cartesian equation

Common Parametric Curves

Circle: $x = r\cos t$, $y = r\sin t$

Ellipse: $x = a\cos t$, $y = b\sin t$

Line: $x = x_0 + at$, $y = y_0 + bt$

Cycloid: $x = r(t - \sin t)$, $y = r(1 - \cos t)$

Derivatives

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}(\frac{dy}{dx})}{dx/dt}$

Arc Length

$L = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$

Area Under Curve

$A = \int_a^b y \frac{dx}{dt} \, dt$

Surface of Revolution

$S = 2\pi \int_a^b y\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$

Sequences & Series

Arithmetic Sequence

$a_n = a_1 + (n-1)d$ where $d$ = common difference

$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$

Geometric Sequence

$a_n = a_1 \cdot r^{n-1}$ where $r$ = common ratio

$S_n = a_1 \cdot \frac{1-r^n}{1-r}$ (if $r \neq 1$)

Infinite Geometric Series

Converges if $|r| < 1$

$S = \frac{a_1}{1-r}$

Sigma Notation

$\sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n$

Series Formulas

$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$

$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{k=1}^{n} k^3 = [\frac{n(n+1)}{2}]^2$

Convergence Tests

Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, diverges

Geometric: Converges if $|r| < 1$

p-Series: $\sum \frac{1}{n^p}$ converges if $p > 1$

Conic Sections

Circle

$(x-h)^2 + (y-k)^2 = r^2$

Center: $(h, k)$, Radius: $r$

Parabola

Vertical: $(x-h)^2 = 4p(y-k)$

Horizontal: $(y-k)^2 = 4p(x-h)$

Vertex: $(h, k)$, Focus: $(h, k+p)$ or $(h+p, k)$

Directrix: $y = k-p$ or $x = h-p$

Ellipse

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

Center: $(h, k)$

$c^2 = |a^2 - b^2|$ (distance from center to foci)

Eccentricity: $e = \frac{c}{a}$, $0 < e < 1$

Hyperbola

$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ (horizontal)

$\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ (vertical)

$c^2 = a^2 + b^2$

Asymptotes: $y - k = \pm\frac{b}{a}(x-h)$

Eccentricity: $e = \frac{c}{a}$, $e > 1$

General Conic

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

Discriminant: $\Delta = B^2 - 4AC$

$\Delta < 0$ (same sign $A,C$): Ellipse

$\Delta = 0$: Parabola

$\Delta > 0$: Hyperbola

Systems & Matrices

Linear System (2×2)

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} e \\ f \end{pmatrix}$

Matrix Operations

Add/Sub: Element by element (same dimensions)

Multiply: Rows of A × Cols of B

Determinant (2×2)

$\det(A) = ad - bc$

Determinant (3×3)

$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

Inverse (2×2)

$A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Cramer's Rule (2×2)

$x = \frac{\det(\begin{pmatrix} e & b \\ f & d \end{pmatrix})}{\det(A)}$

$y = \frac{\det(\begin{pmatrix} a & e \\ c & f \end{pmatrix})}{\det(A)}$

Row Echelon Form

Gaussian elimination for solving systems

1. Pivot on leading entry

2. Eliminate below pivot

3. Move to next row

$\det(A) \neq 0$: Unique solution, $A^{-1}$ exists

Complex Numbers

Standard Form

$z = a + bi$ where $i = \sqrt{-1}$ and $i^2 = -1$

$a$ = real part, $b$ = imaginary part

Operations

$(a+bi) + (c+di) = (a+c) + (b+d)i$

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Conjugate & Modulus

Conjugate: $\overline{z} = a - bi$

Modulus: $|z| = \sqrt{a^2 + b^2}$

$z \cdot \overline{z} = |z|^2$

Division

$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}$

Polar Form

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

$r = |z| = \sqrt{a^2+b^2}$

$\theta = \arg(z) = \tan^{-1}(\frac{b}{a})$

De Moivre's Theorem

$[r(\cos\theta + i\sin\theta)]^n = r^n(\cos(n\theta) + i\sin(n\theta))$

Roots

$\sqrt[n]{z} = \sqrt[n]{r}(\cos(\frac{\theta + 2\pi k}{n}) + i\sin(\frac{\theta + 2\pi k}{n}))$

for $k = 0, 1, 2, \ldots, n-1$

Vectors

Vector Basics

$\mathbf{v} = \langle a, b \rangle = a\mathbf{i} + b\mathbf{j}$ (2D)

$\mathbf{v} = \langle a, b, c \rangle$ (3D)

Magnitude & Direction

$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$ or $\sqrt{a^2 + b^2 + c^2}$

Direction angle: $\theta = \tan^{-1}(\frac{b}{a})$

Unit Vector

$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$

Vector Operations

$\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$

$k\mathbf{v} = \langle ka, kb \rangle$ (scalar multiplication)

Dot Product

$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$

$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta$

Orthogonality & Projection

Orthogonal if $\mathbf{u} \cdot \mathbf{v} = 0$

$\text{comp}_\mathbf{u} \mathbf{v} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|}$ (scalar component)

$\text{proj}_\mathbf{u} \mathbf{v} = (\text{comp}_\mathbf{u} \mathbf{v})\frac{\mathbf{u}}{\|\mathbf{u}\|}$

Cross Product (3D)

$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$

$\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin\theta$ (area)

Limits Preview (Calculus)

Limit Definition

$\lim_{x \to c} f(x) = L$ means $f(x)$ gets arbitrarily close to $L$ as $x$ approaches $c$

One-Sided Limits

Left limit: $\lim_{x \to c^-} f(x)$ (from left)

Right limit: $\lim_{x \to c^+} f(x)$ (from right)

Limit exists if left = right

Limit Laws

$\lim_{x \to c} [f(x) + g(x)] = L + M$

$\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$

$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ if $M \neq 0$

Special Limits

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Continuity

$f$ is continuous at $c$ if:

1. $f(c)$ is defined

2. $\lim_{x \to c} f(x)$ exists

3. $\lim_{x \to c} f(x) = f(c)$

Infinite Limits

$\lim_{x \to \infty} \frac{1}{x} = 0$

$\lim_{x \to \infty} \frac{P(x)}{Q(x)} = \frac{\text{leading coeff of P}}{\text{leading coeff of Q}}$ (if same degree)

Limits are the foundation of calculus—derivatives and integrals build on this concept!

PRE-CALCULUS Complete Formula Sheet